Neither of the two techniques, alone, is powerful enough to solve all tractable CSPs, even in the non-promise variant and on Boolean domain.
In fact, the elusive interaction between consistency-checking methods and linear equations was the major obstacle to the proof of the Feder-Vardi dichotomy conjecture [1]},
finally settled in [2]}, [3]}.
Hence, efforts have been directed to blending the two techniques, in order to design a stronger local-global algorithm [4]}, [5]}.
In [5]}, Brakensiek, Guruswami, Wrochna, and Živný proposed an algorithm that combines the first level of the Sherali-Adams LP hierarchy (known as the basic linear programming (\(\operatorname{BLP}\) ) relaxation) with the \(\operatorname{AIP}\)  relaxation, and characterised its power. Remarkably, that algorithm, which we call \(\operatorname{BA}\)  in this paper, solves all tractable cases of Schaefer's dichotomy of Boolean CSPs [7]}.
This has led some researchers to believe that the algorithmic hierarchy based on \(\operatorname{BA}\)  could be a universal constraint-satisfaction solver – i.e., a constant level of the hierarchy could solve all tractable CSPs, cf. [5]}, [9]}, [10]}, [11]}.
